Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups

Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.


Introduction
Starting with the celebrated work by Lott-Villani [29] and Sturm [34,35], recent years have seen a lot of research interest in extending the notion of Ricci curvature, or more precisely lower Ricci curvature bounds, beyond the realm of classical differential geometry to spaces with singularities [2][3][4]16], discrete spaces [17,30,31] or even settings where there is no underlying space at all as for example in noncommutative geometry [11,12,15,26,32,38,39].
Most of these approaches take as their starting point either the characterization of lower Ricci curvature bound in terms of convexity properties of the entropy on Wasserstein space [37] or in terms of Bakry-Émery's Γ 2 -criterion [6], which derives from Bochner's formula, and in many settings, these two approaches yield equivalent or at least closely related notions of lower Ricci curvature bounds.
One of the reasons to seek to extend the notion of Ricci curvature beyond Riemannian manifolds is that lower Ricci curvature bounds have strong geometric consequences and are a powerful tool in proving functional inequalities. This motivated the investigation of lower Ricci curvature bounds in the noncommutative setting, or for quantum Markov semigroups.
From a positive noncommutative lower Ricci curvature bound in terms of the Γ 2 -condition, Junge and Zeng [21,22] derived a L p -Poincaré-type inequality and transportation inequalities, and under such non-negative lower Ricci curvature bounds Junge and Mei proved L p -boundedness of Riesz transform [20]. Following Lott-Sturm-Villani, Carlen and Maas [10][11][12] studied the noncommutative lower Ricci curvature bound via the geodesic semi-convexity of entropy by introducing a noncommutative analog of the 2-Wasserstein metric. The similar approach was carried out by the first-named author in the infinitedimensional setting in [38]. These two notions of lower Ricci curvature bounds are in general different, but they can both be characterized in terms of a gradient estimate [12,38,39]. A stronger notion of lower Ricci curvature bound, which implies the bound in terms of Γ 2 -condition and in terms of transportation, was introduced by Li, Junge and LaRacuente [26]. See also the further work of Li [25], and Brannan, Gao and Junge [7,8].
However, for many applications in geometric consequences such as the Bonnet-Myers theorem, and functional inequalities such as the concavity of entropy power, a lower bound on the Ricci curvature is not sufficient, but one needs an upper bound on the dimension as well. This leads to the curvaturedimension condition, whose noncommutative analog will be the main object of this article. As a finite-dimensional analog of lower Ricci curvature bounds, the curvature-dimension condition also admits various characterizations. Similar to the "infinite-dimensional" setting, two main approaches describing curvaturedimension conditions are Γ 2 -criterion following Bakry-Émery and convexity properties of entropy on the 2-Wasserstein space in the spirit of Lott-Sturm-Villani. For metric measure spaces, the equivalence of various characterizations on curvature-dimension conditions and their applications have been extensively studied beginning with [16].
While the notion of dimension is built into the definition of manifolds, it is not obvious in the extended settings and requires new definitions. The goal of this article is to provide such a definition of dimension (upper bounds) in the context of quantum Markov semigroups in a way that it fits well with the previously developed notions of lower Ricci curvature bounds in this framework. This definition allows us to prove interesting consequences on the geometry of the state space as well as some functional inequalities.
Furthermore, for quantum Markov semigroups satisfying an intertwining condition, which already appeared in [11] and subsequent work, we provide an easily verifiable upper bound on the dimension, namely the number of partial derivatives in the Lindblad form of the generator. This sufficient condition enables us to prove the curvature-dimension condition in various concrete examples such as quantum Markov semigroups of Schur multipliers and semigroups generated by conditionally negative definite length functions on group algebras.
It should be mentioned that a notion of dimension for a quantum diffusion semigroup already appeared implicitly in the work of König and Smith on the quantum entropy power inequality [24]. In particular, from their entropy power inequality one may also derive the concavity of entropy power for the associated quantum diffusion semigroup. See [1,14,19] for more related work. This example fits conceptually well with our framework as it satisfies the intertwining condition and the dimension in the entropy power considered there is the number of partial derivatives in the Lindblad form of the generator, although the semigroup acts on an infinite-dimensional algebra and is therefore not covered by our finite-dimensional setting. Here we consider the concavity of the entropy power for arbitrary symmetric quantum Markov semigroups over matrix algebras.
In this paper we will focus on two noncommutative analogues of curvaturedimension conditions: the Bakry-Émery curvature dimension condition BE(K, N ), formulated via the Γ 2 -condition, and the gradient estimate GE(K, N ), which is in the spirit of Lott-Sturm-Villani when the reference operator mean is chosen to be the logarithmic mean. They are generalizations of "infinite-dimensional" notions BE(K, ∞) and GE(K, ∞) in previous work, but let us address one difference in the "finite-dimensional" setting, i.e. N < ∞. As we mentioned above, in the "infinite-dimensional" case, i.e. N = ∞, GE(K, ∞) recovers BE(K, ∞) if the operator mean is the left/right trivial mean. However, this is not the case when N < ∞; BE(K, N ) is stronger than GE(K, N ) for the left/right trivial mean.
This article is organized as follows. Section 2 collects preliminaries about quantum Markov semigroups and noncommutative differential calculus that are needed for this paper. In Sect. 3 we study the noncommutative Bakry-Emery curvature-dimension condition BE(K, N ), its applications and the complete version. In Sect. 4 we investigate the noncommutative gradient estimate GE(K, N ) for arbitrary operator means, give an equivalent formulation in the spirit of the Γ 2 -criterion, and also introduce their complete form. Section 5 is devoted to the gradient estimate GE(K, N ), its connection to the geodesic (K, N )-convexity of the (relative) entropy and applications to dimensiondependent functional inequalities. In Sect. 6 we give some examples of quantum Markov semigroups for which our main results apply. In Sect. 7 we discuss how to extend the theory from this article to quantum Markov semigroups that are not necessarily tracially symmetric and explain the main challenge in this case.

Quantum Markov Semigroups and Noncommutative Differential Calculus
In this section we give some background material on quantum Markov semigroups, their generators, first-order differential calculus and operator means.

Quantum Markov Semigroups
Throughout this article we fix a finite-dimensional von Neumann algebra M with a faithful tracial state τ . By the representation theory of finite-dimensional C * -algebras, M is of the form n j=1 M kj (C) and τ = n j=1 α j tr M k j (C) with α j ≥ 0, n j=1 α j k j = 1. Here M n (C) denotes the full n-by-n matrix algebra and tr Mn(C) is the usual trace over M n (C).
Denote by M + the set of positive semi-definite matrices in M. A density matrix is a positive element ρ ∈ M with τ (ρ) = 1. The set of all density matrices is denoted by S(M) and the set of all invertible density matrices by S + (M). We write L 2 (M, τ) for the Hilbert space obtained by equipping M with the inner product The adjoint of a linear operator T : M → M with respect to this inner product is denoted by T † . We write id for the identity operator, with an index indicating on which space it acts if necessary.
A family It is the unique linear operator on M such that P t = e −tL . Let us remark that sign conventions differ and sometimes −L is called the generator of (P t ). Let σ ∈ S + (M). The quantum Markov semigroup (P t ) is said to satisfy the σ-detailed balance condition (σ-DBC) if τ (P t (x)yσ) = τ (xP t (y)σ) for x, y ∈ M and t ≥ 0. In the special case σ = 1 we say that (P t ) is tracially symmetric or symmetric, and denote A tracially symmetric quantum Markov semigroup (P t ) is ergodic if 1 is the unique invariant state of (P t ).
Although it is not necessary to formulate the curvature-dimension conditions, we will deal exclusively with tracially symmetric quantum Markov semigroups since all examples where we can verify the conditions fall into that class. As a special case of Alicki's theorem [5,Theorem 3] (see also [11, where J is a finite index set, ∂ j = [v j , · ] for some v j ∈ M, and for every j ∈ J there exists a unique j * ∈ J such that v * j = v j * . We call the operators ∂ j partial derivatives. Using the derivation operator ∂ := (∂ j ) j∈J : M →M := ⊕ j∈J M, we may also write L = ∂ † ∂. Let L(ρ) and R(ρ) be the left and right multiplication operators on L 2 (M, τ), respectively, and fix an operator mean Λ. For ρ ∈ M + we definê ρ = Λ(L(ρ), R(ρ)).

Noncommutative Differential Calculus and Operator Means
Of particular interest for us are the cases when Λ is the logarithmic mean Λ log (L(ρ), R(ρ)) = With Λ = Λ log being the logarithmic mean, we have the chain rule identity for log (see [11,Lemma 5.5] for a proof): Here and in what follows, we use the notation ρ(x 1 , . . . , x n ) := (ρx 1 , . . . ,ρx n ).

Bakry-Émery Curvature-Dimension Condition BE(K, N )
This section is devoted to the noncommutative analog of the Bakry-Émery curvature-dimension condition BE(K, N ) defined by the Γ 2 -criterion. After giving the definition, we will show that it is satisfied for certain generators in Lindblad form, where the dimension parameter N is given by the number of partial derivatives. We will then prove that BE(K, N ) implies an improved Poincaré inequality. In the final part of this section we study a complete version of BE(K, N ), called CBE(K, N ), and show that it has the expected tensorization properties. As usual, we write Γ(a) for Γ(a, a) and Γ 2 (a) for Γ 2 (a, a).

Bakry-Émery
for any a ∈ M: If this is the case, we say the semigroup (P t ) satisfies Bakry-Émery curvaturedimension condition BE(K, N ).
Proof. The proof is essentially based on the following identities: For s ∈ [0, t], which follow by direct computations. To prove (a) =⇒ (b), we set To show (b) =⇒ (a), we put for any t > 0: By the Kadison-Schwarz inequality, for any unital completely positive map Φ on M and b ∈ M. If we apply this to P s and use the assumption from (b), we get which proves (a).  We shall give a sufficient condition for Bakry-Émery curvature-dimension condition BE(K, N ). Before that we need a simple inequality.
Proof. In fact, Definition 3.6. Suppose that L is the generator of the tracially symmetric quantum Markov semigroup (P t ) with the Lindblad form:

Proposition 3.7. Suppose that the generator L of the tracially symmetric quantum Markov semigroup (P t ) admits the Lindblad form (LB). Then for any a,
Proof. Note that . This, together with the Leibniz rule for ∂ j 's (so also ∂ † j 's), and the fact that So by definition, the carré du champ operator is given by: The above computations yield and where in the last equality we used again the fact that {∂ j } = {∂ † j }. This proves (3.1). If (P t ) satisfies the K-intertwining condition, then

Moreover, by Lemma 3.5 we get
Therefore (P t ) satisfies BE(K, d):

Applications
In this subsection we present two applications of the Bakry-Émery curvaturedimension condition, namely a Poincaré inequality and a Bonnet-Myers theorem.
It is well-known that when K > 0, the dimensionless bound BE(K, ∞) implies that the smallest non-zero eigenvalue of the generator is at least K. As a simple application of the dimensional variant we show that this bound can be improved. N ) and λ 1 is the smallest non-zero eigenvalue of L, then Proof. By BE(K, N ) we have In particular, if La = λ 1 a and a 2 = 1, then from which the desired inequality follows.
To state the Bonnet-Myers theorem, we recall the definition of the metric d Γ on the space of density matrices that is variously known as quantum L 1 -Wasserstein distance, Connes distance or spectral distance. It is given by for ρ 0 , ρ 1 ∈ S(M). Proposition 3.9. Let K, N ∈ (0, ∞). If a symmetric quantum Markov semigroup (P t ) is ergodic and satisfies Bakry-Émery curvature-dimension condition BE(K, N ), then Proof. The proof follows the same line as that of [28,Theorem 2.4]. The condition BE(K, N ) implies Thus for any ρ ∈ S(M), Since (P t ) is assumed to be ergodic, we have P t ρ → 1 as t → ∞, and we end up with Remark 3.10. If (P t ) is not ergodic, then the same argument gives

Complete BE(K, N )
In many applications it is desirable to have estimates that are tensor-stable in the sense that they hold not only for (P t ), but also for (P t ⊗ id Mn(C) ) with a constant independent of n ∈ N, as this allows to analyze complex composite systems by studying their subsystems separately. Even in the case N = ∞, it seems to be unknown if this tensor stability holds for the Bakry-Émery estimate. For that reason we introduce the complete Bakry-Émery estimate CBE(K, N ), which has this tensor stability by definition. We will show that this stronger estimate also holds for quantum Markov semigroups satisfying the K-intertwining condition, and moreover, this estimate behaves as expected under arbitrary tensor products. Definition 3.11. Let K ∈ R and N > 0. We say that the quantum Markov semigroup for all x 1 , . . . , x n ∈ M and t > 0.
Just as in Proposition 3.1 one can show that CBE(K, N ) is equivalent to for all x 1 , . . . , x n ∈ M and t ≥ 0. For N = ∞, this criterion was introduced in [21] for group von Neumann algebras under the name algebraic Γ 2 -condition.
To show that CBE(K, N ) for (P t ) is equivalent to BE(K, N ) for (P t ⊗ id Mn(C) ) with constants independent of n, we need the following elementary lemma.
A direct computation shows In the following two results we will give two classes of examples for which the condition CBE is satisfied. Proposition 3.14. Suppose that the generator L of the quantum Markov semigroup (P t ) admits the Lindblad form (LB) with d partial derivatives ∂ 1 , . . . , ∂ d . If (P t ) satisfies the K-intertwining condition for K ∈ R, then (P t ) satisfies CBE(K, d).
Proof. A direct computation shows that L ⊗ id Mn(C) admits a Lindblad form with partial derivatives ∂ 1 ⊗id Mn(C) , . . . , ∂ d ⊗id Mn(C) . Now the claim is a direct consequence of Propositions 3.7 and 3.13. Proof. By assumption, M ∼ = C(X) for a compact space X. We have to show for any α j ∈ C.
Before we state the tensorization property of CBE, we need another elementary inequality.

Lemma 3.16. Let
A be a C*-algebra. If a, b ∈ A and λ > 0, then Proof. In fact,  N ) and Proof. We use superscripts for the (iterated) carré du champ to indicate the associated quantum Markov semigroup. Let κ = min{K, K }. We have By CBE(κ, N ) for (P t ) and CBE(κ, N ) for (Q t ) we have Moreover, by Lemma 3.16, which shows BE(κ, N +N ) for (P t ⊗Q t ). To prove CBE(κ, N + N ), we can simply apply the same argument to (P t ⊗ id Mn(C) ) and (Q t ⊗ id Mn(C) ) for arbitrary n ∈ N. GE(K, N )

Gradient Estimate GE(K, N ) and a Sufficient Condition
In [10][11][12]38], a noncommutative analog of the 2-Wasserstein metric was constructed on the set of quantum states. Among other things, it gives rise to a notion of (entropic) lower Ricci curvature bound via geodesic semi-convexity of the entropy. This allows to prove a number of functional inequalities under strictly positive lower Ricci curvature bound, including the modified log-Sobolev inequality that (seemingly) cannot be produced under the Bakry-Emery curvature-dimension condition BE(K, ∞).
This entropic lower Ricci curvature bound is captured in the following gradient estimate or equivalently Re ∂La,ρ∂a + 1 2 where the notationsρ and · ρ correspond to the logarithmic mean Λ log . Recall Sect. 2 for more details. The fact that logarithmic mean comes into play lies in the use of chain rulê In fact, for the gradient estimate (GE(K, ∞)) and its equivalent form 4.1 one can work with any operator mean. This not only makes the theory more flexible, but also includes the Bakry-Émery curvature-dimension condition BE(K, ∞) as a special case. Indeed, one recovers BE(K, ∞) by replacing the logarithmic mean in (4.1) with the left/right trivial mean. In the next section we discuss the connection between GE(K, N ) and (K, N )-convexity of the (relative) entropy.
The study of (GE(K, ∞)) for arbitrary operator means was started in [38,39]. Here we continue to work within this framework and focus on the "finite-dimensional" version of (GE(K, ∞)) or (4.1), which we call gradient estimate GE(K, N ).

Definition 4.1. Let Λ be an operator mean and (P t ) be a symmetric quantum
Markov semigroup whose generator takes the Lindblad form (LB). We say that (P t ) satisfies the gradient estimate GE(K, for any t ≥ 0, a ∈ M and ρ ∈ S + (M).

Remark 4.2.
Both sides of (GE(K, N )) make sense for arbitrary ρ ∈ S(M) and are continuous in ρ. Thus, if ρ ∈ S(M) is not invertible, one can apply (GE(K, N )) to ρ = ρ+ 1 1+ , which is invertible for > 0, and let 0 to see that still holds for any t ≥ 0 and a ∈ M.  Proof. Assume that (P t ) satisfies (GE (K, N )). Set Then φ(t) ≥ 0 and φ(0) = 0. Therefore φ (0) ≥ 0, that is, This is nothing but (4.2), since dG(ρ)(Lρ) = − d dt t=0 P t ρ. Now suppose that (P t ) satisfies 4.2. Fix t > 0 and put Psρ , 0 ≤ s ≤ t. Then applying (4.2) to (ρ, a) = (P s ρ, P t−s a), we get This, together with the fundamental theorem of calculus, yields Therefore (P t ) satisfies (GE (K, N )).  In the case N = ∞, the gradient estimate GE(K, ∞) for the left trivial mean is equivalent to the exponential form of BE(K, ∞). For N < ∞ this seems to be no longer the case, but one still has one implication: the Bakry-Emery curvature-dimension condition BE(K, N ) is stronger than GE(K, N ) for the left trivial mean. This is a consequence of Cauchy-Schwarz inequality for the state τ (ρ · ): Similar to BE(K, N ), the intertwining condition is also sufficient to prove GE(K, N ) with the same dimension (upper bound).
Theorem 4.7. Let (P t ) be a symmetric quantum Markov semigroup over M with the Lindblad form (LB). Suppose that (P t ) satisfies K-intertwining condition for some K ∈ R. Then for any operator mean Λ the quantum Markov semigroup (P t ) satisfies GE(K, d).

Proof. For a ∈ M, recall that
Under the K-intertwining condition, we have (either by Kadison-Schwarz or BE(K, ∞)) P s Γ(P t−s a) ≥ e 2Ks Γ(P t a).
By (3.2) and Lemma 3.5, we get for any ( From K-intertwining condition and Cauchy-Schwarz inequality for the state τ (ρ·) on M, this is bounded from below by So we have proved that for any x ∈M: Replacing x by x * , we obtain Note that the second summand is the same in both cases. Now since Λ is an operator mean, we have where in the first inequality we used the monotonicity, concavity (Lemma 2.1 (b)) and positive homogeneity (Lemma 2.1 (a)) of Λ, and in the second inequality we used the transformer inequality and Lemma 2.1(d). This, together with the K-intertwining condition, yields ∂P t a 2 ρ = Λ(L(ρ), R(ρ))∂P t a, ∂P t a = e −2Kt P t Λ(L(ρ), R(ρ))P t ∂a, ∂a This completes the proof, by Remark 4.5.

Bonnet-Myers Theorem
As a first application of the dimensional gradient estimate GE (K, N ), we present here a Bonnet-Myers theorem for the noncommutative analog of the Wasserstein distance introduced in [11,12]. The proof is quite similar (or, in fact, similar to the dual) to the proof of Proposition 3.9.
Let us first recall the definition of the metric. The space S + (M) of invertible density matrices is a smooth manifold and the tangent space at ρ ∈ S + (M) can be canonically identified with the traceless self-adjoint elements of M. Assume that (P t ) is an ergodic tracially symmetric quantum Markov semigroup with generator L with Lindblad form (LB).

Complete GE(K, N )
Now we turn to the complete version of GE(K, N ). In this part we always fix an operator mean Λ.

Definition 4.10.
We say that a quantum Markov semigroup (P t ) satisfies complete gradient estimate CGE(K, N ) for K ∈ R and N ∈ (0, ∞] if (P t ⊗id Mn(C) ) satisfies GE(K, N ) for all n ∈ N (for the fixed operator mean Λ).
Similar to Proposition 3.14, the K-intertwining condition is sufficient for CGE : Proposition 4.11. Suppose that the generator L of the quantum Markov semigroup (P t ) admits the Lindblad form (LB) with d partial derivatives ∂ 1 , . . . , ∂ d . If (P t ) satisfies the K-intertwining condition for K ∈ R, then (P t ) satisfies CGE (K, d).
Also, the complete gradient estimate CGE is tensor stable.

Proposition 4.12. Consider two quantum Markov semigroups
Proof. For each j = 1, 2, we denote by L j the generator of (P j t ) and ∂ j : M j → M j (to distinguish from partial derivatives ∂ j 's) the corresponding derivation operator so that L j = (∂ j ) † ∂ j . Denote P t = P 1 t ⊗ P 2 t . Then its generator is L = ∂ † ∂, where the derivation operator ∂ is given by Since (P j t ) satisfies CGE(K, N j ), j = 1, 2, we have for any a ∈M := ⊗ jMj and ρ ∈ S + (M) that As we have proven in [39,Theorem 4.1], for the first summand one has As for the second summand, note that L = L 1 ⊗ id + id ⊗ L 2 . So by Cauchy-Schwarz inequality, All combined, we obtain

Geodesic (K, N )-Convexity of the (Relative) Entropy and Relation to the Gradient Estimate GE(K, N )
In the case of the logarithmic mean, the given quantum Markov semigroup is the gradient flow of the (relative) entropy with respect to the transport distance W. In this case, the gradient estimate GE(K, ∞) is equivalent to geodesic K-convexity of the (relative) entropy with respect to W, and several functional inequalities can be obtained using gradient flow techniques. Similarly, the gradient estimate GE(K, N ) is equivalent to geodesic (K, N )-convexity of the (relative) entropy with respect to W, a notion introduced by Erbar, Kuwada and Sturm [16], and again, gradient flow techniques allow to deduce several dimensional functional inequalities from the abstract theory of (K, N )-convex functions on Riemannian manifolds.

(K, N )-Convexity for the (Relative) Entropy
Let (M, g) be a Riemannian manifold and K ∈ R, N ∈ (0, ∞]. A function for all x ∈ M and v ∈ T x M . With the function the (K, N )-convexity of S can equivalently be characterized by For N = ∞, one obtains the usual notion of K-convexity. Moreover, the notion of (K, N )-convexity is obviously monotone in the parameters K and N in the sense that if S is (K, N )-convex, then S is also (K , N )-convex for K ≤ K and N ≥ N . Our focus will be on the case when F is the (relative) entropy and the Riemannian metric is the one introduced in [11,12], whose definition was recalled in Sect. 4.2.
If F : S + (M) → R is smooth, its Fréchet derivative can be written as for a unique traceless self-adjoint x ∈ M. This element x shall be denoted by DF (ρ). In particular, if F (ρ) = τ (ρ log ρ), then DF (ρ) = log ρ + c for some c ∈ R. By [11,Theorem 7.5], the gradient of F is given by (recall (4.5) for K Λ ρ ) Of particular interest to us is the case when F is the (relative) entropy, that is, the functional Ent : S + (M) → (0, ∞), Ent(ρ) = τ (ρ log ρ).
If we choose Λ to be the logarithmic mean Λ log , then ρ t = P t ρ satisfies the gradient flow equationρ t = −∇ g Λ Ent(ρ t ) for any ρ ∈ S + (M) [11,Theorem 7.6]. For this reason, we fix the operator mean Λ to be the logarithmic mean in this section.
To formulate the metric formulations of (K, N )-convexity, we need the following notation: For θ, κ ∈ R and t ∈ [0, 1] put sκ(θ) , κθ 2 = 0 and κθ 2 < π 2 , t, κθ 2 = 0, +∞, The following theorem is a quite direct consequence of the abstract theory of (K, N )-convex functions and the computation of the gradient and Hessian on (S + (M), g) carried out in [11,12]. Nonetheless, it implies some interesting functional inequalities, as we shall see in the following subsection.

Dimension-Dependent Functional Inequalities
Let us first collect some consequences of (K, N )-convexity that were already observed in [16], adapted to our setting. Recall that Ent(ρ) = τ (ρ log ρ). We use the notation for the Fisher information.
It satisfies the de Bruijn identity d dt Ent(P t ρ) = −I(P t ρ).

Concavity of Entropy Power
Let us now move on to the concavity of entropy power: For the heat semigroup on R n , the concavity of entropy power along the heat flow was first proved by Costa in [13]. In [36] Villani gave a short proof and remarked that this can be proved using Γ 2 -calculus. Recently Li and Li [27] considered this problem on the Riemannian manifold under the curvaturedimension condition CD(K, N ). Here we show that the geodesic concavity of the entropy power follows from the (K, N )-convexity of the entropy. N ) for the logarithmic mean, then This implies the concavity of the entropy power t → U N (P t ρ) 2 .
Proof. Let ρ t = P t ρ. Since Ent is (K, N )-convex by Theorem 5.1 and (P t ) is a gradient flow of Ent in (S + (M), g) by our choice of the operator mean, we have Remark 5.4. The same proof applies abstractly whenever F is a (K, N )-convex functional on a Riemannian manifold and (ρ t ) is a gradient flow curve of F .
The following proof is closer to the spirit of Villani.
Another proof of Theorem 5.3. Put ϕ(t) := − Ent(ρ t ) = −τ (ρ t log ρ t ) with ρ t = P t ρ. Recall the chain rule ∂ρ =ρ∂ log ρ. Thus This, together with (5.2) and (5.5), yields So by (5.6) we get Remark 5.5. Here we used the fact that I = II, or equivalently, If we consider the heat semigroup P t = e tΔ on R n , then this follows from the elementary identity as used in Villani's proof [36].

Examples
In this section we present several classes of examples of quantum Markov semigroups satisfying CBE(K, N ) and CGE(K, N ). The verification of these examples relies crucially on the criteria from Propositions 3.14 and 4.11.

Schur Multipliers Over Matrix Algebras
A Schur multiplier A over the n × n matrix algebra M n (C) is a linear map of the form: where a ij ∈ C and {e ij } n i,j=1 are the matrix units. By Schoenberg's theorem (see for example [9, Appendix D]), α i α j a ij ≤ 0, whenever α 1 , . . . , α n are complex numbers such that n j=1 α j = 0. If this is the case, then there exists a real Hilbert space H and elements a(j) ∈ H, 1 ≤ j ≤ n, such that Suppose that (e k ) 1≤k≤d is an orthonormal basis of H. Define for each 1 ≤ k ≤ d v k := n j=1 a(j), e k e jj ∈ M n (C).
Then for any 1 ≤ i, j ≤ n: By the choice of (e k ), we have Therefore, and it is easy to see that [v k , A·] = A[v k , ·] for each k. So by Propositions 3.14 and 4.11 we have CBE(0, d) and CGE(0, d) for any operator mean.

Herz-Schur Multipliers Over Group Algebras
Let G be a finite group. Suppose that λ is the left-regular representation, i.e. for g ∈ G, where 1 h is the delta function at h. The group algebra of G is then the (complex) linear span of {λ g | g ∈ G}, denoted by C[G]. It carries a canonical tracial state τ given by τ (x) = x1 e , 1 e , where e is the unit element of G.
We say that : G → [0, ∞) is a conditionally negative definite length function if (e) = 0, (g −1 ) = (g) for all g ∈ G and g,h∈G are complex numbers such that g∈G α g = 0. By Schoenberg's theorem (see for example [ Every conditionally negative definite length function on G induces a τsymmetric quantum Markov semigroup (P t ) on C[G] characterized by P t λ g = e −t (g) λ g for g ∈ G. Let e 1 , . . . , e d be an orthonormal basis of H. As argued in [39] (or similar to the Schur multipliers case), the generator L of (P t ) can be written as The operators v j are not contained in C[G] in general, but one can extend L to a linear operator on B( 2 (G)) by the same formula, and a direct computation shows [v j , L · ] = L[v j , · ]. By Propositions 3.14 and 4.11, (P t ) satisfies CBE(0, d) and CGE(0, d) for any operator mean.
Example 6.1. The cyclic group Z n = {0, 1, . . . , n − 1}; see [21,Example 5.9] or [39,Example 5.7]: Its group (von Neumann) algebra is spanned by λ k , 0 ≤ k ≤ n − 1. One can embed Z n to Z 2n , so let us assume that n is even. The word length of k ∈ Z n is given by (k) = min{k, n − k}. The associated 1-cocycle can be chosen with H = R n 2 and b : Z n → R where (e j ) 1≤j≤ n 2 is an orthonormal basis of R n 2 . Thus the quantum Markov semigroup associated with satisfies CBE(0, n/2) and CGE(0, n/2) for any operator mean.

Generalized Depolarizing Semigroups
It is finite for any E and can be computed explicitly in terms of the multiplicities of ran(E) inside of M d (C). In the special case when E(a) = τ (a)1, we have C(E) = d and C cb (E) = d 2 .
The generalized depolarizing semigroup (or dephasing semigroup) associated with E is given by Let L = id − E be the generator of (P t ) with Lindblad form Fix k ∈ J . Since LE = 0, we have Choosing Λ as the right-trivial mean and not applying Cauchy-Schwarz, one obtains in a similar manner τ (Γ(P t a)ρ) ≤ e −t τ (Γ(a)P t ρ) − 1 − e −t C(E) τ (|La| 2 P t ρ).

Curvature-Dimension Conditions Without Assuming Tracial Symmetry
In plenty of applications one encounters quantum Markov semigroups that are not necessarily tracially symmetric, but only satisfy the detailed balance condition σ-DBC (with σ = 1) we mentioned in Sect. 2. Many of the results from this article still apply in this case, with one important caveat, as we will discuss here. The definition of the Bakry-Émery gradient estimate BE(K, N ) makes sense for arbitrary quantum Markov semigroups on matrix algebras without any change, and all the consequences we proved stay valid in this more general setting with essentially the same proofs.
The gradient estimate GE(K, N ) relies on the Lindblad form of the generator of the semigroup. By Alicki's theorem, a similar Lindblad form exists for generators of quantum Markov semigroups satisfying the σ-DBC, and the norms ξ ρ have been defined in this setting in [11,12] -in fact, instead of a single operator mean one can choose a family of operator connections that depends on the index j. With this norm, one can formulate GE(K, N ) as ∂P t a 2 ρ ≤ e −2Kt ∂a 2 where one now has to distinguish between P t and P † t because of the lack of tracial symmetry.
The connection between a generalization of the metric W, the semigroup (P t ) and the relative entropy still remains true in this more general setting with an appropriate modification of the definition of W [11,12], so that the identification of GE(K, N ) with the (K, N )-convexity condition for an entropy functional from Theorem 5.1 along with its applications also has an appropriate analog for quantum Markov semigroups satisfying the σ-DBC.
However, the criteria from Proposition 3.7 and Theorem 4.7, which actually allow us to verify BE(K, N ) and GE(K, N ) in concrete examples, rely crucially on the Lindblad form of generators of tracially symmetric quantum Markov semigroups and do not immediately carry over to the σ-detailed balance case. Thus the question of proving BE(K, N ) and GE(K, N ) for not necessarily tracially symmetric quantum Markov semigroups remains open, so its usefulness in this case is still to be proven.